Math 1051, Robertson, Exam 3 on Chapters 3 & 4 on Friday 12 November 2010 KEY page 1
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1 Math, Robertson, Eam on Chapters & on Friday November 0 KEY page. You earned points out of. Ans: f 6 Write the equation of a quadratic function whose graph has the following characteristics: It opens down; it is stretched by a factor of ; it s ais of symmetry is the line = ; it passes through the point (, -). f a h k y k When, y k 6 k 6 f. You earned points out of. Ans: A Epress the area A of an equilateral triangle as a function of where is the length of one side of the triangle. A bh h To find the height, use Pythagoras: c a b h h h h A bh h. You earned points out of. Ans: ARC = Find the Average Rate of Change of f from to. The ARC of a linear function is the slope of the line. In this case, m. - -
2 Math, Robertson, Eam on Chapters & on Friday November 0 KEY page. You earned points out of. Given the function f a. [] At what point does the graph cross the -ais? Ans: Cross: (-, 0) The zero at = - has odd multiplicity so the graph crosses the -ais at (-, 0). b. [] At what point does the function touch the -ais? Ans: Touch: (, 0) The zero at = has even multiplicity so the graph touches the -ais at (, 0). c. [8] What is the behavior of the function near each of its zeros? f look like That is, what function does near each zero? Ans: Near left zero f 9 Ans: Near left zero f Near = - the function looks like - 9 This is a very steep reflected cubic shape. f. Near = the function looks like f This is a very steep upside down parabolic shape
3 Math, Robertson, Eam on Chapters & on Friday November 0 KEY page. You earned points out of 0. Given the function f : a. [8] Write the function in verte form. Ans: f b. [] What are the coordinates of the verte? Ans: Verte:,8 We can read the verte directly from the equation:,8. Or, the -coordinate is given by b verte a f 8 To find the y-coordinate, calculate f f verte c. [] What is the equation of the ais of symmetry. Ans: Ais of symmetry: = The ais of symmetry is the vertical line that passes through the verte. The equation of any vertical line is = k, where k is a constant. In this case k is the value of verte. So, the equation of the ais of symmetry is =. d. [] What are the coordinates of the -intercepts? Ans: -intercepts:, 0,,0 The -intercepts are the values of when y = or e. [] What are the coordinates of the y-intercept? Ans: y-intercept: 0, The y-intercept is the value of y when = 0: f y 0 y
4 Math, Robertson, Eam on Chapters & on Friday November 0 KEY page 6. You earned points out of. Given the function f 9 : a. [] What are the equations of the vertical asymptotes? Ans: = - and = - The function reduces to g. The denominator is 0 when = - and = -. So, the vertical asymptotes are the lines = - and = -. b. [] What are the coordinates of the -intercepts? Ans: (, 0) and (-, 0) c. [] What are the coordinates of any holes in the graph, if any? Ans: 0, The common factor of cancels so there is a hole when = 0. The y-coordinate is g So, the hole is at 0 0 0,. g 0. d. [] What is the equation of the horizontal or oblique asymptote? Ans: y The degree of the numerator and denominator is the same so there is a horizontal asymptote whose value is the ratio of the coefficients of the dominant terms. This is the line y
5 Math, Robertson, Eam on Chapters & on Friday November 0 KEY page 7. You earned points out of 0. On the grid on the net page, sketch the graph of the function f 6 On your graph, be sure to clearly show asymptotes, holes, intercepts, and indicate where the graph crosses an asymptote if it does.. Factor, state domain, and THEN reduce. f g There is a hole when - = 0, which means when = The y-coordinate of the hole is g So, the hole is at,.. Plot intercepts. For -intercepts, state whether graph touches (multiplicity is even) or crosses (multiplicity is odd) -ais. -intercepts at = 0 and = -6. Both have multiplicity odd so graph crosses at (0, 0) and (-6, 0).. Draw vertical asymptotes (the zeros of the denominator). The denominator of the reduced fraction is 0 when = - so = - is a VA.. Draw horizontal asymptotes: deg num = deg denom +, so use long division to find oblique asymptote, y = m + b 6 g So, the OA is y = Plot where the graph crosses a horizontal or oblique asymptote, if it does. g y This has no solution so the graph does not cross the OA..
6 Math, Robertson, Eam on Chapters & on Friday November 0 KEY page 6 6. If needed, plot a few etra points. 7. Connect the dots You earned points out of. Ans:,,0 Solve: Write your answer using interval notation.. Write inequality with 0 on one side Find the real zeros of the numerator and denominator. = - and = 0. Use the real zeros to break up the number line into intervals. - - f g h - Interval : 0. Use a test point in each interval to see if the function is pos or neg. Interval : 0 Try / // Get f / Simplify : f 0 yes yes no We want f 0 so,,0. Note that f(-) = 0 so we must eclude -.
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